Question

A map in a pirate's log gives directions to the location of a buried treasure. The starting location is an old oak tree. According to the map, the treasure's location is found by proceeding 20 paces north from the oak tree and then 30 paces northwest. At this location, an iron pin is sunk in the ground. From the iron pin, walk 10 paces south and dig. How far (in paces) from the oak tree is the spot at which digging occurs?

Short answer

Answer: The digging spot is approximately 29.15 paces away from the old oak tree.

Step-by-step solution

Represent the displacement vectors

First, let's represent the three displacements as vectors using the information in the problem: - From the oak tree to the iron pin: 20 paces north + 30 paces northwest - From the iron pin to the digging spot: 10 paces south Note: 1 pace north is represented as (1,0), 1 pace south as (-1,0), 1 pace east as (0,1), and 1 pace west as (0,-1). For northwest, imagine splitting the pace in equal parts i.e half north and half west. So, 1 pace northwest would be (0.5,-0.5).

Calculate the vectors

Now, we will find the vectors for each displacement. The vector for 20 paces north: 20 * (1,0) = (20,0) The vector for 30 paces northwest: 30 * (0.5,-0.5) = (15,-15) The vector for 10 paces south: 10 * (-1,0) = (-10,0)

Add the displacement vectors

Now, add the first two vectors (oak tree to iron pin) to get the total displacement from the oak tree to the iron pin, and then add the vector from the iron pin to the digging spot: vector_oak_to_iron_pin: (20,0) + (15,-15) = (35,-15) vector_iron_pin_to_digging_spot: (-10,0) vector_oak_to_digging_spot: (35,-15) + (-10,0) = (25,-15)

Find the magnitude of the total displacement vector

Now, to find the distance from the oak tree to the digging spot, we will use the Pythagorean theorem to find the magnitude of the vector: distance = sqrt((25)^2 + (-15)^2) = sqrt(625 + 225) = sqrt(850)

Final Answer

So, the spot where digging occurs is approximately sqrt(850) ≈ 29.15 paces away from the old oak tree.

Key concepts

Vector Addition

Vector addition is a fundamental concept in physics that enables us to combine several displacements or forces into a single resultant vector. When a pirate follows a map to a treasure, they are essentially performing vector addition in real life by moving from one point to another along various paths.

For the math behind this, imagine each pace as an arrow that points in the direction one should move. In vector addition, these arrows—or vectors—are placed tip-to-tail. The final position relative to the starting point is found by drawing a vector from the start of the first vector to the end of the last vector. This resultant vector gives both the direction and the distance of the displacement.

In our pirate's treasure problem, the movement from the oak tree to the iron pin and then to the digging spot involves adding together several vectors that represent paces in different directions—north, northwest, and south.

Pythagorean Theorem

The Pythagorean theorem is an essential concept in understanding the magnitude of vectors, especially when they form a right triangle. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as the formula: \( c^{2} = a^{2} + b^{2} \), where \(c\) is the hypotenuse and \(a\) and \(b\) are the two other sides.

In our problem, the magnitude of the total displacement vector is analogous to finding the hypotenuse. The individual east/west and north/south components represent the other two sides of the triangle. By applying the Pythagorean theorem to the components of the displacement vectors, we can solve for the total distance—the 'paces'—the pirate must travel to reach the treasure.

Magnitude of Vectors

The magnitude of a vector—often thought of as the 'length' of the vector—represents the distance from the starting point to the ending point in a straight line, irrespective of the path taken. In a two-dimensional space, if we have a vector with components \(x\) and \(y\), we calculate the magnitude as \(\sqrt{x^2 + y^2}\) using the Pythagorean theorem.

In our example, the magnitude of the displacement vectors provides the answer to how far the pirate must dig from the original oak tree. After vector addition, the final displacement vector's components yield the necessary inputs to compute this magnitude, delivering the total distance in paces.

Physics Problem-Solving

Physics problem-solving often involves breaking down a complex movement into simpler parts called vectors, calculating those vectors, and then reassembling them to find a result. This step-by-step approach is crucial to systematically solve problems in a logical and comprehensible way.

In the case of the pirate's treasure hunt, this methodical approach means identifying each leg of the journey as a separate vector, adding those vectors together, and finally calculating the magnitude of the resultant vector. This orderly process converts a real-world scenario into a solvable physics problem, providing a clear methodology for understanding and computing distances and displacements.